Question: Simplify the following expression: $y = \dfrac{-3x^2- 5x+12}{-3x + 4}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-3)}{(12)} &=& -36 \\ {a} + {b} &=& &=& {-5} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-36$ and add them together. Remember, since $-36$ is negative, one of the factors must be negative. The factors that add up to ${-5}$ will be your ${a}$ and ${b}$ When ${a}$ is ${4}$ and ${b}$ is ${-9}$ $ \begin{eqnarray} {ab} &=& ({4})({-9}) &=& -36 \\ {a} + {b} &=& {4} + {-9} &=& -5 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-3}x^2 +{4}x) + ({-9}x +{12}) $ Factor out the common factors: $ x(-3x + 4) + 3(-3x + 4)$ Now factor out $(-3x + 4)$ $ (-3x + 4)(x + 3)$ The original expression can therefore be written: $ \dfrac{(-3x + 4)(x + 3)}{-3x + 4}$ We are dividing by $-3x + 4$ , so $-3x + 4 \neq 0$ Therefore, $x \neq \frac{4}{3}$ This leaves us with $x + 3; x \neq \frac{4}{3}$.